Normalization of a delta function in curved spacetime

I think the first one is incorrect in curved spacetime, or in general when the metric is non-constant. I would argue this by saying that the delta function does not transform, whereas the fourth-order differential transforms in the opposite way to [itex]\sqrt{-g}[/itex], so the whole thing transforms as a scalar as it must.

I've also heard that [itex]\delta^4[/itex] is not a scalar, which suggests that (1) is the correct statement. However, this seems strange to me as I would think that (1) will fail to hold in curvilinear coordinates e.g.

I think the first one is incorrect in curved spacetime, or in general when the metric is non-constant. I would argue this by saying that the delta function does not transform, whereas the fourth-order differential transforms in the opposite way to [itex]\sqrt{-g}[/itex], so the whole thing transforms as a scalar as it must.

I've also heard that [itex]\delta^4[/itex] is not a scalar, which suggests that (1) is the correct statement. However, this seems strange to me as I would think that (1) will fail to hold in curvilinear coordinates e.g.

I'd say that

[itex]\int d^4 x \delta^4(x - x_0) = 1[/itex]

is the usual definition. The RHS is trivially a scalar. The measure on the LHS is a density. So the delta distribution is also a density, as was mentioned by others here.